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An optimal matching problem for the Euclidean distance. (English) Zbl 1297.49006

Based on quotations from the authors’ introduction: In this paper, the authors are interested in an optimal matching problem that consists in transporting two commodities to a prescribed location, the target set, in such a way that they match there and the total cost of the operation, which is measured in terms of the Euclidean distance that the commodities are transported, is minimized. Optimal matching problems for uniformly convex cost were analyzed by many authors and have implications in economic theory. However, when one considers the Euclidean distance as cost, new difficulties appear since one deals with a non-uniformly convex cost. The optimal matching problem under consideration is related to the classical Monge-Kantorovich mass transport problem. By using tools from this theory, the existence of a solution of the optimal matching problem follows. The existence of a solution is true for any norm in \(\mathbb R^N\). The authors show the existence of a matching measure concentrated on the boundary of the target set. The main contribution in this paper is to demonstrate a method to solve the problem taking the limit as \(p\to\infty\) in a system of PDEs of \(p\)-Laplacian type, which allows to give more information about the matching measure and the Kantorovich potentials for the involved transport. The limit as \(p\to\infty\) in the system requires some care since the system is nontrivially coupled and therefore the estimates for one component are related to the ones for the other.

MSC:

49J20 Existence theories for optimal control problems involving partial differential equations
49Q20 Variational problems in a geometric measure-theoretic setting
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
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